94 \clearpage |
94 \clearpage |
95 |
95 |
96 %----------// T O C \\----------% |
96 %----------// T O C \\----------% |
97 |
97 |
98 \pagenumbering{Roman} |
98 \pagenumbering{Roman} |
99 This thesis is structured into a fundamental part introducing the thesis aswell as the {\sisac{}} project and describing the mathematic base. Further a automatically generated practical part representing the work on {\sisac{}} which can be extended. |
99 %This thesis is structured into a fundamental part introducing the motivation, the basic notions concerning the thesis aswell as the {\sisac{}} project and describing the mathematic base. Further a automatically generated practical part representing the work on {\sisac{}} which can be extended. |
100 \tableofcontents |
100 \tableofcontents |
101 \clearpage |
101 \clearpage |
102 \pagenumbering{arabic} |
102 \pagenumbering{arabic} |
103 \setcounter{page}{6} |
103 \setcounter{page}{6} |
104 |
104 |
105 %----------// PART-1 \\----------% |
105 %----------// PART-1 \\----------% |
106 |
106 |
107 \part{Project Fundamentals} |
107 \part{Project Fundamentals} |
108 |
108 |
109 The goals of the thesis are finally defined in section \ref{sec:goals} which seems to be very late. The reason for this fact is that there has a lot of research to be done prior and a lot of this research has to be described in this thesis before we are able to define the proper goals. All this is neccessary for understanding the effort on this work. |
109 %The goals of the thesis are finally defined in section \ref{sec:goals} which seems to be very late. The reason for this fact is that there has a lot of research to be done prior and a lot of this research has to be described in this thesis before we are able to define the proper goals. All this is neccessary for understanding the effort on this work. |
110 |
110 |
111 \section{Introduction} |
111 \section{Introduction} |
112 The motivation to this thesis mainly takes it source from the feeling of understanding difficult signal processing tasks and the will to help others to get this feeling to. |
112 %The motivation to this thesis mainly takes it source from the feeling of understanding difficult signal processing tasks and the will to help others to get this feeling to. |
113 \par Signal Processing (SP) requieres a huge range of mathematic knowledge as well as a feeling for simplification and number tricks but even though this fact, the operations themself are no higher ones. The main task is to understand. Aside this description we think of the classic math ideas and techniques, consisting of predefined formulas, notations and forumularsations we learn. |
113 %\par Signal Processing (SP) requieres a huge range of mathematic knowledge as well as a feeling for simplification and number tricks but even though this fact, the operations themself are no higher ones. The main task is to understand. Aside this description we think of the classic math ideas and techniques, consisting of predefined formulas, notations and forumularsations we learn. |
114 |
114 % |
115 \subsection{Mechanization of Mathematics} |
115 Didactics of mathematics faces a specific issue, a gap between (1) introduction of math concepts and skills and (2) application of these concepts and skills, which ususally are separated into different units in curricula (for good reasons). For instance, (1) teaching partial fraction decomposition is separated from (2) application for inverse Z-transform in signal processing. |
116 A problem behind is the mechanization of mathematic theories in CTP-bases languages. There is still a hugh gap between these theories and this what we call an applications - in Example Signal Processing. |
116 |
117 \begin{example} |
117 This gap is an obstacle for applying math as an foundamental thinking technology in engineering: In (1) motivation is lacking because the question ``What is this stuff good for~?'' cannot be treated sufficiently, and in (2) the ``stuff'' is not available to students in higher semesters as widespread experience shows. |
118 \[ |
118 |
119 X\cdot(a+b)+Y\cdot(c+d)=aX+bX+cY+dY |
119 \medskip |
120 \] |
120 Motivated by this didactical issue on the one hand, and ongoing R\&D on a novel kind of educational mathematics assistant at Graz University of Technology~\footnote{http://www.ist.tugraz.at/isac/} promising to cope with this issue on the other hand, several institutes are planning to join their expertise: the Institutes for Institute for Information Systems and Computer Media (IICM), the Institute for Software Technology (IST), the Institutes for Mathematics, the Signal Processing and Speech Communication Laboratory (SPSC Lab), the Institute for Structural Analysis and the Institute of Electrical Measurement and Measurement Signal Processing. |
121 {\small\textit{ |
121 |
122 \noindent A very simple example on this what we call gap is the simplification above. It is needles to say that it is correct and also isabell forfills it correct - \emph{always}. But sometimes we don't want do simplificate these things, sometimes it is easyer for handling and understanding if we keep terms together. Think of a problem were we now would need only the coefficients of $X$ and $Y$. This is what we call the gap between applications and theorem proofment. |
122 This thesis is the first attempt to tackle the above mentioned issue, it focuses on Telematics, because these specific studies focus on mathematics in STEOP, the introductory orientation phase. STEOP is considered an opportunity to investigate the impact of {\sisac}'s prototype on the issue and others. |
123 }} |
123 |
124 \caption{Correct but not usefull}\label{eg:gap} |
124 \medskip |
125 \end{example} |
125 The thesis is structured as follows: Part I concerns theory, part II the implementation work, where the latter is the main part. |
126 Until we are not able to fill this gap we have to live with it but first have a look on the meaning of this statement: |
126 |
127 \par Mechanized math starts from mathematical models and \emph{hopefully} proceeds to match physics. Academic engineering starts from physics (experimentation, measurement) and then proceeds to mathematical modelling and formalization. The process from a physical observance to a mathematical theory is unavoidable bound of setting up a big collection of standards, rules, definition but also exceptions. These are the things making mechanization that difficult. |
127 In part I, \S\ref{isabisac} gives a brief description of the state-of-the-art for educational math assistants (\S\ref{emas}) and introduces the notions required for the implementation work (\S\ref{math-auth}). In particular, \S\ref{user-guid} explains, why math authoring in \sisac{} is {\em not} concerned with interaction (and thus not with user guidance etc at all~!). So a concise description of the thesis' goals needs to be postponed to \S\ref{sec:goals}. |
128 \begin{example} |
128 |
129 \[ |
129 \S\ref{sp} analyzes ten (TODO: exact no?) problems defined by the SPSC Lab for the knowledge already provided (\S\ref{know-isab}, \S\ref{know-isac}), discusses the selection of problems for implementation (\S\ref{know-missing}) TODO: further structure ? |
130 m,\ kg,\ s,\ldots |
130 %(\S\ref{}) |
131 \] |
131 |
132 {\small\textit{ |
132 \section{Mechanization of Math in Isabelle/{\isac}}\label{isabisac} |
133 \noindent Think about some units like that one's above. Behind each unit there is a discerning and very accurate definition: One Meter is the distance the light travels, in a vacuum, through the time of 1 / 299.792.458 second; one kilogramm is the weight of a platinum-iridium cylindar in paris; and so on. But are these definitions useable in a computer mechanized world?! |
133 %A problem behind is the mechanization of mathematic theories in CTP-bases languages. There is still a hugh gap between these theories and this what we call an applications - in Example Signal Processing. |
134 }} |
134 %\begin{example} |
135 \caption{Units in measurement}\label{eg:units} |
135 % \[ |
136 \end{example} |
136 % X\cdot(a+b)+Y\cdot(c+d)=aX+bX+cY+dY |
137 \par A computer or a CTP-System builds on programms witth predefined logical ruels and does not know any mathematical trick (follow up example \ref{eg:trick}) or recipe to walk around difficult expressions. |
137 % \] |
138 \begin{example} |
138 % {\small\textit{ |
139 \[ \frac{1}{j\omega}\cdot\left(e^{-j\omega}-e^{j3\omega}\right)= \] |
139 % \noindent A very simple example on this what we call gap is the simplification above. It is needles to say that it is correct and also isabell forfills it correct - \emph{always}. But sometimes we don't want do simplificate these things, sometimes it is easyer for handling and understanding if we keep terms together. Think of a problem were we now would need only the coefficients of $X$ and $Y$. This is what we call the gap between applications and theorem proofment. |
140 \[ \frac{1}{j\omega}\cdot e^{-j2\omega}\cdot\left(e^{j\omega}-e^{-j\omega}\right)= |
140 % }} |
141 \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$\frac{1}{j}\,\left(e^{j\omega}-e^{-j\omega}\right)$}= \] |
141 % \caption{Correct but not usefull}\label{eg:gap} |
142 \[ \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$2\, sin(\omega)$} \] |
142 %\end{example} |
143 {\small\textit{ |
143 %Until we are not able to fill this gap we have to live with it but first have a look on the meaning of this statement: |
144 \noindent Sometimes it is also usefull to be able to apply some \emph{tricks} to get a beautiful and particulary meaningful result, which we are able to interpret. But as seen in this example it can be hard to find out what operations have to be done to transform a result into a meaningful one. |
144 %\par Mechanized math starts from mathematical models and \emph{hopefully} proceeds to match physics. Academic engineering starts from physics (experimentation, measurement) and then proceeds to mathematical modelling and formalization. The process from a physical observance to a mathematical theory is unavoidable bound of setting up a big collection of standards, rules, definition but also exceptions. These are the things making mechanization that difficult. |
145 }} |
145 %\begin{example} |
146 \caption{Mathematic tricks}\label{eg:trick} |
146 % \[ |
147 \end{example} |
147 % m,\ kg,\ s,\ldots |
148 For such a system the only possibility is to work through its known definitions and stops if none of these fits. Specified on Signal Processing or any other application it is often possible to walk through by doing simple creases. This creases are in generell based on simple math operatiopms but the challange is to teach the machine \emph{all}\footnote{Its pride to call it \emph{all}.} of them. Unfortunataly the goal of CTP Isabelle is to reach a high level of \emph{all} but it in real it will still be a survey of knowledge which links to other knowledge and {\sisac{}} a trainer and helper but no human compensating calulator. |
148 % \] |
149 \par {\sisac{}} itselfs aims to adds an \emph{application} axis (formal specifications of problems outof topics from Signal Processing, etc.) and an \emph{algorithmic} axis to the \emph{deductive} axis of physical knowledge. The result is a three-dimensional universe of mathematics. |
149 % {\small\textit{ |
150 |
150 % \noindent Think about some units like that one's above. Behind each unit there is a discerning and very accurate definition: One Meter is the distance the light travels, in a vacuum, through the time of 1 / 299.792.458 second; one kilogramm is the weight of a platinum-iridium cylindar in paris; and so on. But are these definitions useable in a computer mechanized world?! |
151 \subsubsection*{Notes on Mechanization of Mathematics} |
151 % }} |
152 This thesis tries to \emph{connect} these two worlds and is one of the first guidelines to implement problem classes in {\sisac}. As we are still in a eary part of development, this is the first thesis dealing within this topic and there is \emph{no} related work to guid through. A more detailed description about this fact can be found in Section \ref{sec:related}. |
152 % \caption{Units in measurement}\label{eg:units} |
153 The major challenge of the practical part, of this thesis, is, that "connecting the two worlds" involves programming in a CTP-based programming language which is in a very early state of prototyping. There is no concrete experience data ready to grep. |
153 %\end{example} |
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154 %\par A computer or a CTP-System builds on programms witth predefined logical ruels and does not know any mathematical trick (follow up example \ref{eg:trick}) or recipe to walk around difficult expressions. |
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155 %\begin{example} |
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156 %\[ \frac{1}{j\omega}\cdot\left(e^{-j\omega}-e^{j3\omega}\right)= \] |
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157 %\[ \frac{1}{j\omega}\cdot e^{-j2\omega}\cdot\left(e^{j\omega}-e^{-j\omega}\right)= |
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158 % \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$\frac{1}{j}\,\left(e^{j\omega}-e^{-j\omega}\right)$}= \] |
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159 %\[ \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$2\, sin(\omega)$} \] |
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160 % {\small\textit{ |
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161 % \noindent Sometimes it is also usefull to be able to apply some \emph{tricks} to get a beautiful and particulary meaningful result, which we are able to interpret. But as seen in this example it can be hard to find out what operations have to be done to transform a result into a meaningful one. |
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162 % }} |
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163 % \caption{Mathematic tricks}\label{eg:trick} |
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164 %\end{example} |
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165 %For such a system the only possibility is to work through its known definitions and stops if none of these fits. Specified on Signal Processing or any other application it is often possible to walk through by doing simple creases. This creases are in generell based on simple math operatiopms but the challange is to teach the machine \emph{all}\footnote{Its pride to call it \emph{all}.} of them. Unfortunataly the goal of CTP Isabelle is to reach a high level of \emph{all} but it in real it will still be a survey of knowledge which links to other knowledge and {\sisac{}} a trainer and helper but no human compensating calulator. |
|
166 %\par {\sisac{}} itselfs aims to adds an \emph{application} axis (formal specifications of problems outof topics from Signal Processing, etc.) and an \emph{algorithmic} axis to the \emph{deductive} axis of physical knowledge. The result is a three-dimensional universe of mathematics. |
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167 % |
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168 %\subsubsection*{Notes on Mechanization of Mathematics} |
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169 %This thesis tries to \emph{connect} these two worlds and is one of the first guidelines to implement problem classes in {\sisac}. As we are still in a eary part of development, this is the first thesis dealing within this topic and there is \emph{no} related work to guid through. A more detailed description about this fact can be found in Section \ref{sec:related}. |
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170 %The major challenge of the practical part, of this thesis, is, that "connecting the two worlds" involves programming in a CTP-based programming language which is in a very early state of prototyping. There is no concrete experience data ready to grep. |
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171 % |
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172 As mentioned in the introduction, a prototype of an educational math assistant called \sisac\footnote{\sisac=\textbf{Isa}belle \cite{Nipkow-Paulson-Wenzel:2002} for \textbf{C}alculations, see http://www.ist.tugraz.at/isac/.} bridges the gap between (1) introducation and (2) application of mathematics: {\sisac} is based on Computer Theorem Proving (CTP), a technology which requires each fact and each action justified by formal logic, so \sisac{} makes justifications transparent to students in interactive step-wise problem solving. By that way {\sisac} already can serve both: |
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173 \begin{enumerate} |
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174 \item Introduction of math stuff (e.g. partial fraction decomposition) by stepwise explaining and exercising respective symbolic calculations with ``next step guidance (NSG)'' and rigorously checking steps freely input by students --- this also in context with advanced applications (where the stuff to be taught in higher semesters can be skimmed through by NSG), and |
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175 \item Application of math stuff in advanced engineering courses (e.g. problems to be solved by inverse Z-transform in a Signal Processing Lab) --- and now without much ado about basic math techniques (like partical fraction decomposition): ``next step guidance'' supports students in independenly (re-)adopting such techniques. |
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176 \end{enumerate} |
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177 Before the question is answers, how {\sisac} accoplishes this task from a technical point of view, some remarks on the state-of-the-art is given: |
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178 |
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179 \subsection{Educational Mathematics Assistants (EMAs)}\label{emas} |
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180 Educational software in mathematics is, if at all, based on Computer Algebra Systems (CAS, for instance \cite{progr-mathematica,prog-maple06}), Dynamic Geometry Systems (DGS, for instance \footnote{GeoGebra http://www.geogebra.org, Cinderella http://www.cinderella.de/, GCLC http://poincare.matf.bg.ac.rs/~janicic/gclc/}) or spread-sheets. These base technologies are used to program math lessons and sometimes even exercises. The latter are cumbersome: the steps towards a solution of such an interactive exercise need to be provided with feedback, where at each step a wide variety of possible input has to be foreseen by the programmer --- so such interactive exercises either require high development efforts or the exercises constrain possible inputs. |
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181 |
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182 A new generation of educational math assistants (EMAs) is emerging presently, which is based on Computer Theorem Proving (CTP). CTP, for instance Isabelle \cite{Nipkow-Paulson-Wenzel:2002} and Coq \cite{Huet_all:94}, is a technology which requires each fact and each action justified by formal logic. Pushed by demands for \textit{proven} correctness of safety-critical software CTP advances into software engineering; from these advancements computer mathematics benefits in general, and math education in particular. Two features of CTP are immediately beneficial for learning: |
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183 |
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184 \paragraph{CTP have knowledge in human readable format,} that is in standard predicate calculus. CTP following the LCF-tradition have that knowledge down to the basic definitions of set, equality, etc~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL.html}; following the typical deductive development of math, natural numbers are defined and their properties proven~\footnote{http://isabelle.in.tum.de/dist/library/HOL/Number_Theory/Primes.html}, etc. Present knowledge mechanized in CTP exceeds high-school mathematics by far, however by knowledge required in software technology, and not in other engineering sciences. |
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185 |
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186 \paragraph{CTP can model the whole problem solving process} in mathematical problem solving {\em within} a coherent logical framework. This is already being done by three projects, by Ralph-Johan Back \cite{Back-SD09}, by ActiveMath \cite{ActiveMath-MAIN11} and by Carnegie Mellon Tutor \cite{mat-tutor-cmu-MAIN11}. |
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187 |
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188 Having the whole problem solving process within a logical coherent system, such a design guarantees correctness of intermediate steps and of the result (which seems essential for math software); and the second advantage is that CTP provides a wealth of theories which can be exploited for mechanizing other features essential for educational software. |
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189 |
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190 \subsection{Generation of User Guidance in EMAs}\label{user-guid} |
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191 One essential feature for educational software is feedback to user input and assistance in coming to a solution. |
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192 |
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193 \paragraph{Checking user input} by ATP during stepwise problem solving is being accomplished by the three projects mentioned above \cite{Back-SD09,ActiveMath-MAIN11,mat-tutor-cmu-MAIN11} exclusively. They model the whole problem solving process as mentioned above, so all what happens between formalized assumptions (or formal specification) and goal (or fulfilled postcondition) can be mechanized. Such mechanization promises to greatly extend the scope of educational software in stepwise problem solving. |
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194 |
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195 \paragraph{Next step guidance (NSG)} comprises the system's ability to propose a next step; this is a challenge for CTP: either a radical restriction of the search space by restriction to very specific problem classes is required, or much care and effort is required in designing possible variants in the process of problem solving \cite{proof-strategies-11}. |
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196 |
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197 Another approach is restricted to problem solving in engineering domains, where a problem is specified by input, precondition, output and postcondition, and where the postcondition is proven by ATP behind the scenes \cite{wn:lucas-interp-12}: Here the possible variants in the process of problem solving are provided with feedback {\em automatically}, if the problem is described in a CTP-based programming language~\cite{plmms10}: the programmer only describes the math algorithm without caring about interaction (the respective program is functional and even has no in/output statements~!); interaction is generated as a side-effect by the interpreter --- an efficient separation of concern between math programmers and dialog designers promising application all over engineering disciplines. |
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198 |
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199 |
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200 \subsection{Math Authoring in Isabelle/\isac}\label{math-auth} |
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201 Authoring new mathematics knowledge in {\sisac} can be compared with ``application programming'' of engineering problems; most of such programming uses CAS-based programming languages (CAS = Computer Algebra Systems; e.g. Mathematica's \cite{progr-mathematica} or Maple's programming language \cite{prog-maple06}). |
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202 |
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203 {\sisac}, however, uses a novel type of CTP-based language \cite{plmms10} for describing how to constuct a solution to an engineering problem and for calling equation solvers, integration, etc~\footnote{Implementation of CAS-like functionality in CTP is not primarily concerned with efficiency, but with a didactic question: What to decide for: for high-brow algorithms at the state-of-the-art or for elementary algorithms comprehensible for students~?} within CTP; CTP can ensure ``systems that never make a mistake'' \cite{casproto} --- are impossible for CAS which have no logics underlying. |
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204 |
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205 With writing such CTP-based programs authoring is perfect, the application programmer is not concerned with interaction or with user guidance: this is concern of a novel kind of program interpreter called Lucas-Interpreter \cite{wn:lucas-interp-12}. This interpreter hands over control to a dialog component at each step of calculation (like a debugger at breakpoints) and calls automated CTP to check user input following personalized strategies according to a feedback module. |
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206 |
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207 \medskip |
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208 However ``application programming with CTP'' is not done with writing a program: according to the principles of CTP, each step must be justified. Such justifications are given by theorems. So all steps must be related to some theorem, if there is no such theorem it must be added to the existing knowledge, which is organized in so-called \textbf{theories} in Isabelle. A theorem must be proven; fortunately Isabelle comprises a mechanism (called ``axiomatization''), which allows to omit proofs. Such a theorem is, for instance %TODO: take your example ! |
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209 {\small\begin{tabbing} |
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210 123\=123\=123\=123\=\kill |
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211 \>axiomatization where \\ |
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212 \>\> rule1: "1 = $\delta$ [n]" and\\ |
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213 \>\> rule2: "|| z || > 1 ==> z / (z - 1) = u [n]" and\\ |
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214 \>\> rule3: "|| z || < 1 ==> z / (z - 1) = -u [-n - 1]" and \\ |
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215 \>\> rule4: "|| z || > || $\alpha$ || ==> z / (z - $\alpha$) = $\alpha^n$ * u [n]" and\\ |
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216 \>\> rule5: "|| z || < || $\alpha$ || ==> z / (z - $\alpha$) = -($\alpha^n$) * u [-n - 1]" and\\ |
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217 \>\> rule6: "|| z || > 1 ==> z/(z - 1)$^2$ = n $\cdot$ u [n]"\\ |
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218 \end{tabbing} |
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219 } |
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220 |
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221 In order to provide CTP with logical facts for checking user input, the Lucas-Interpreter requires a \textbf{specification}. Such a specification is for instance |
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222 {\small\begin{tabbing} |
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223 123,\=postcond \=: \= $\forall \,A^\prime\, u^\prime \,v^\prime.\,$\=\kill |
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224 Specification no.1:\\ |
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225 %\>input\>: $\{\;r={\it arbitraryFix}\;\}$ \\ |
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226 \>input \>: $\{\;r\;\}$ \\ |
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227 \>precond \>: $0 < r$ \\ |
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228 \>output \>: $\{\;A,\; u,v\;\}$ \\ |
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229 \>postcond \>:{\small $\;A=2uv-u^2 \;\land\; (\frac{u}{2})^2+(\frac{v}{2})^2=r^2 \;\land$}\\ |
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230 \> \>\>{\small $\;\forall \;A^\prime\; u^\prime \;v^\prime.\;(A^\prime=2u^\prime v^\prime-(u^\prime)^2 \land |
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231 (\frac{u^\prime}{2})^2+(\frac{v^\prime}{2})^2=r^2) \Longrightarrow A^\prime \leq A$} \\ |
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232 \>props\>: $\{\;A=2uv-u^2,\;(\frac{u}{2})^2+(\frac{v}{2})^2=r^2\;\}$ |
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233 \end{tabbing} |
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234 } |
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235 Such a specification is checked before the execution of a program is started, the same applies for sub-programs. In the following example program the sub-programs are designated by {\tt SubProblem}: TODO one example. |
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236 |
154 |
237 |
155 \subsection{Goals of the Thesis}\label{sec:goals} |
238 \subsection{Goals of the Thesis}\label{sec:goals} |
156 Imagine a piece of software would be able to support you by understanding every problem class, upcoming in the first years attending university - wouldn't it be great? |
239 %Imagine a piece of software would be able to support you by understanding every problem class, upcoming in the first years attending university - wouldn't it be great? |
157 \par {\sisac{}} tries to do that, but the current state of the art is miles away from this goal and a single implementation of a problem is not enough to cahnge this circumstamce. Through this fact it is all the more essential to try, test, research and document the implementation of problem classes from "`real world"' applications. Responding to the abstract at the begin of this document the thesis has two folds; on the one hand certainly to provide interactiv course material for Signal Processing (which means to implement a single problem provided by the Institute of Signal Processing and Speech Communication (SPSC); follow up Calulcations), and to extract experience data respectively help the {\sisac{}}-team by setting up a detailed description of technicalities hacking {\sisac{}} on the other hand. |
240 %\par {\sisac{}} tries to do that, but the current state of the art is miles away from this goal and a single implementation of a problem is not enough to cahnge this circumstamce. Through this fact it is all the more essential to try, test, research and document the implementation of problem classes from "`real world"' applications. Responding to the abstract at the begin of this document the thesis has two folds; on the one hand certainly to provide interactiv course material for Signal Processing (which means to implement a single problem provided by the Institute of Signal Processing and Speech Communication (SPSC); follow up Calulcations), and to extract experience data respectively help the {\sisac{}}-team by setting up a detailed description of technicalities hacking {\sisac{}} on the other hand. |
158 \par Another goal is to demonstrate the power and attractivity of {\sisac}. |
241 %\par Another goal is to demonstrate the power and attractivity of {\sisac}. |
159 |
242 Now all the notions are in place to describe the task ``Interactive Course Material for Signal Processing based on Isabelle/\sisac'' appropriately by the following points: |
160 \section{Mechanization of Signal Processing Problems} |
243 \begin{enumerate} |
161 \subsection{Relevant Knowledge available in Isabelle} |
244 \item Analyze the problems given by the SPSC Lab for mathematics \textbf{knowledge required}, search the knowledge already available in Isabelle/\sisac, estimate efforts required to fill the gap between knowledge required and knowledge available, and finally select problems for implementation accordingly. |
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245 \item Implement the selected problems in Isabelle/\sisac, which means, in appropriate Isabelle theories \textbf{for each problem} implement: |
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246 \begin{enumerate} |
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247 \item \textbf{Definitions and theorems} required within the specification (including ``descriptions'' for input variables and output variables) and the program (proofs omitted via ``axiomaization'') |
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248 \item \textbf{A specification} which describes the input variables, the preconditions on the input (a challenge for rigorously exact mathematics~!), the output variables and the postcondition, which relates input to output such that the problem is characterized formally (another challenge for rigorously exact mathematics~!) |
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249 \item \textbf{A program} describing the algorithm which solves the problem, i.e. which constructs output meeting the postcondition. Programming involves identifying the steps (tactics~!) which create the calculation and calling CAS-functions (simplification, equation solvers, etc) appropriately. Modularization of programs into {\tt SubProblems} has to prepare for re-use of code. |
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250 \end{enumerate} |
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251 \item Add \textbf{multimedia explanations} to each problem (i.e. to specific definitions, theorems, the specification and the program) such that non-expert students (e.g. within STEOP, the introductory orientation phase at TUG) get an idea the problem is about. |
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252 \item \textbf{Document the implementation} such that |
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253 \begin{enumerate} |
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254 % \item Interactive course material hopefully useful in education within the SPSC Lab and within STEOP, the introductory orientation phase at TUG, as a preview for students in Telematics on later application of math knowledge introduced in the first semester and |
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255 % \item A detailed description of technicalities in programming implemented as an interactive Isabelle/Isar theory, providing future programmers with guidelines and {\sisac}-developers with feedback in usability of the CTP-based program language. |
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256 \item subsequent application programmers have guidelines for further implementation of interactive course material in SPSC and other engineering sciences |
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257 \item \sisac{} developers get feedback for ongoing improvement of the CTP-based programming language, the respective development environment and the respective program interpreter (called Lucas-Interpreter) |
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258 \item development of knowledge for engineering sciences is being motivated in the Isabelle community. |
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259 \end{enumerate} |
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260 \end{enumerate} |
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261 |
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262 |
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263 \section{Mechanization of Signal Processing Problems}\label{sp} |
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264 \subsection{Relevant Knowledge available in Isabelle}\label{know-isab} |
162 Isabelle is developed now for a long time and so we are able to access a huge range of theories and usefull snipsets. The main problem according this snipsets is that isabelle still is a theorem proofer and not an algebra system. But due the work of the {\sisac}-development team there are already also many calculation examples provided. |
265 Isabelle is developed now for a long time and so we are able to access a huge range of theories and usefull snipsets. The main problem according this snipsets is that isabelle still is a theorem proofer and not an algebra system. But due the work of the {\sisac}-development team there are already also many calculation examples provided. |
163 \par The SPSC provided a list of problems which are often done wrong or are missunderstood by studentsin term of the problem classes. Out of these tasks we tried to extract the core operations and looked up which parts are already implemented or usefull. The provided problems are: |
266 \par The SPSC provided a list of problems which are often done wrong or are missunderstood by studentsin term of the problem classes. Out of these tasks we tried to extract the core operations and looked up which parts are already implemented or usefull. The provided problems are: |
164 \begin{itemize} |
267 \begin{itemize} |
165 \item Fourier-Transformation |
268 \item Fourier-Transformation |
166 \item Convolution |
269 \item Convolution |